I have two independent random variables A and B with known continuous
density and distribution functions and their pdf's overlap. How do I
calculate the probability that variable A will be lower than variable B?

In an election two candidates, Atif and Bryan, have in a ballot box a
and b votes repectively with a > b, for example 3 and 2. If ballots
are randomly drawn and tallied, what is the chance that at least once
after the first tally the candidates have the same number of tallies?

A discussion of a seeming paradox involving two people comparing
amounts of money in their wallets with the greater amount being given
to the holder of the lesser amount. At first glance it seems that
the game favors both participants, but a closer look reveals that is
not really the case.

Scientists were studying the population of puffins in Alaska. They
spotted and tagged 52 puffins in December. Two months later, they
spotted 50 puffins. Of those 50 puffins, 3/10 of them had been tagged
in December. Predict the population of puffins in Alaska.

Anne, Bob, and Carmel are going to take turns rolling a fair die, in
the order Anne, Bob, Carmel, Anne, Bob Carmel, etc. The first person
to roll a 6 will win $100.
a) Find the probability that Anne will win $100 if the first four
numbers rolled are 3, 2, 5, 5.
b) Find the probability that Anne will win $100 if a draw is to be
declared with nobody winning $100 in the event of a 6 not being
rolled within the first 8 rolls in total.
c) Suppose that a draw is to be declared if ever two 1's are rolled
in a row (by two different players). Also suppose that the game is
in fact over and did not end in a draw. Find the probability that
Anne is the player who won $100.

Problem: If I have n shuffled cards in k colours numbered from 0 to
(n/k)-1, what is the probability that no card will have the same number
on it as its position in the deck modulo n/k? Standard example: n = 52, k
= 4, cards numbered from 0 to 12. A card with number 3 mustn't be on
positions 3, 16, 29, or 42.

Event A happens constantly every 3 seconds, and each time it happens
there is a 15% chance it triggers Event B. Event B lasts 12 seconds.
Given that Event B would restart if it's re-triggered while already in
effect, what is the average time Event B will run when it is started?

Can you use a coin (which has 2 events of equal probability) to devise
three events with 1/3rd probability each? I find devising 4 events
easy; toss the coin twice and interpret the results 00, 01, 10, 11
(i.e. HH, HT, TH, TT) as the 4 events. But, is it okay to disregard
any one result (say 00) and claim that the other three events are of
equal probability? I find it hard to believe that a coin can be used
to have three events of equal probability.

Each of 16 prisoners receives a hat that is either red or blue (the
colour is selected randomly; each has a 1/2 probability). All the
prisoners must simultaneously either try to guess the colour of their
hats, or pass.

80 percent of light bulbs last 2400 hours, 20 percent last 2400 hours...
Given a collection of screws with a Gaussian distribution of size.... The
frequency of a mistake for wires is once in 25 meters...